To test the adaptation mechanism (REF ), we have considered a square lattice of side length 1, with \(N=25\times 25\)  nodes. The positions of the interior nodes of the lattice have been randomly perturbed with a Gaussian noise, with standard deviation \(\sigma =0.5\) . Then, a Delaunay triangulation of the lattice sites is performed, resulting the graph network \({\cal G}\) . The edge lengths \(L_{ij}\)  are then obtained.
We have considered \(K=3\)  sources and \(R=4\)  sinks, randomly allocated to the nodes of the network \({\cal G}\) . Two sources have intensities \(S_{source}=0.4\)  and a third source has intensity \(S_{source}=0.2\) . The sinks have equal intensities \(S_{sink}=-0.25\) . The adaptation process has been calculated with equations (REF ), for the parameter values \(\beta =1\)  and \(\gamma =2/3\) , using the Euler explicit numerical method with time increment \(\Delta \tau =0.1\) . (A simple calculation shows that the discretised Euler method preserves the volume of the channels in the graph, simplifying numerical calculations.) Two different sets of initial conductivities have been considered: homogeneous conductivities all over the edges, \(D_{ij}(0)=1\) , and random conductivities uniformly distributed in the interval \([1/2,3/2]\) . In Figure REF , we show the resulting adapted steady state networks. The steady state graphs in the figure are different but can be considered metrically close. The lengths of the steady state trees are, in a), \(L=2.936\) , and, in b), \(L=2.859\) . For comparison, an upper bound for the length of the Steiner tree connecting sources and sinks is \(L_{us}=2.333\) , [1]}.
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